It was determined via triangulation that the celestial bodies are about the same height as the sun. We have documentation of our method of triangulation, but it is mostly in specific regards to the sun. It was found that the celestial bodies behave similarly, and so they were lumped into the same altitude of the sun. For specifics of the triangulation method, look for the article on the Sun's Distance on the Wiki on the front page.
What evidence is there that the moon is 250,000 miles from the earth?
Common knowledge. This was done many years before radio waves. There is ample evidence of the distance. We are speaking of real evidence. Recent measurements by laser beams. Operations of the amateur radio operators. The distance is not questioned in the real world.
The only thing that flat earth ever tries to contradict this is their old word, which is "denial."
Just one more flat earth fallacy. The earth is a globe. Denial is not going to change reality.
The ancient measurements were made assuming the earth was flat. But the earth is not flat.
But after all, this is The Flat Earth Society and what can you expect from flat earth ?